Título: The infinitesimal generator of the stochastic Burgers equation
Palestrante: Massimiliano Gubinelli, University of Bonn
Data: 09/03/2021
Horário: 13:30h
Local: Transmissão online
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ID da reunião: 958 0581 3232
Resumo: In this talk I will discuss the martingale problem formulation of the Burgers equation (or KPZ equation) in one dimension or similar equations (multicomponent, fractional variants and other hydrodynamic models). The well posedness involves a detailed study of the space of test functions for the generator and of the solution of the associated Kolmogorov backward equation.
Joint work with Nicolas Perkowski and Mattia Turra. Based on the papers Massimiliano Gubinelli and Nicolas Perkowski, The Infinitesimal Generator of the Stochastic Burgers Equation’, Probability Theory and Related Fields 178, no. 3: 1067–1124, click HERE to read. M. Gubinelli and M. Turra, ‘Hyperviscous Stochastic Navier–Stokes Equations with White Noise Invariant Measure’, Stochastics and Dynamics 20, no. 06: 2040005, click HERE to read.
Título: "Weak convergence for the scaled cover time of the rooted binary tree"
Palestrante: Santiago Saglietti
Data: 03/03/2021
Horário: 11h às 12h
Local: Transmissão Online
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Resumo: We consider a continuous time random walk on the rooted binary tree of depth n with all transition rates equal to one and study its cover time, namely the time until all vertices of the tree have been visited. We prove that, normalized by 2^{n+1}n and then centered by (log2)n-log n, the cover time admits a weak limit as the depth of the tree tends to infinity. The limiting distribution is identified as that of a randomly shifted Gumbel random variable with rate one, where the shift is given by the unique solution to a specific distributional equation. The existence of the limit and its overall form were conjectured in the literature. However, our approach is different from those taken in earlier works on this subject and relies in great part on a comparison with the extremal landscape of the discrete Gaussian free field. Joint work with Aser Cortines and Oren Louidor.
Título: Full overlaps in mixing Markov Chains
Palestrante: Rodrigo Lambert (UFU)
Data: 22/02/2021
Horário: 15h - 16h (Rio de Janeiro local time)
Local: Transmissão online
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Resumo: For a string of length n, the overlapping function defines the greatest size of a repetition, in the sense that it is k if its first and last k symbols coincide. When the source that generates the strings satisfies the complete grammar condition, the overlapping function is always non-negative. In the present work we deal with the case where the complete grammar condition is removed, and therefore “negative overlaps” are allowed. We state a weak convergence theorem when the source is a beta-mixing Markov Chain with finite diameter (greatest “distance” between two symbols of the alphabet). This is a work in progress in collaboration with Erika Alejandra Rada-Mora (UFABC).
Todas as palestras são realizadas em inglês.
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Em relação a este ano, alguns dias após cada encontro o vídeo deverá estar disponível AQUI.
Título: Gaussian random permutations and the boson point process
Palestrante: Ines Armendariz (UBA)
Data: 01/03/2021
Horário: 15h - 16h (Rio de Janeiro local time)
Local: Transmissão online
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Resumo: We construct an infinite volume spatial random permutation associated to a Gaussian Hamiltonian, which is parametrized by the point density and the temperature. Spatial random permutations are naturally related to boson systems through a representation originally due to Feynman (1953). Bose-Einstein condensation occurs for dimensions 3 or larger, above a critical density, and is manifest in this representation by the presence of cycles of macroscopic length. For subcritical densities we define the spatial random permutation as a Poisson process of finite unrooted loops of a random walk with Gaussian increments that we call Gaussian loop soup, analogous to the Brownian loop soup of Lawler and Werner (2004). We also construct Gaussian random interlacements, a Poisson process of doubly-infinite trajectories of random walks with Gaussian increments analogous to the Brownian random interlacements of Sznitman (2010). For dimensions greater than or equal to 3 and supercritical densities, we define the spatial permutation as the superposition of independent realizations of the Gaussian loop soup at critical density and Gaussian random interlacements at the remaining density. We show some properties of these spatial permutations, in particular that the point marginal is the boson point process, for any point density.
Título: Conditional propagation of chaos for systems of interacting neurons with random synaptic weights
Palestrante: Eva Löcherbach (Université Paris I)
Data: 08/02/2021
Horário: 15:00 até 16:00h (Horário do Rio de Janeiro)
Local: Transmissão on-line
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Resumo: We study the stochastic system of interacting neurons introduced in De Masi et al 2015, in a diffusive scaling. The system consists of N neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the potential of the spiking neuron is reset to 0 and all other neurons receive an additional amount of potential which is a centred random variable of order 1 / \sqrt (N). In between successive spikes, each neuron's potential follows a deterministic flow. We prove the convergence of the system, as the number of neurons tends to infinity, to a limit nonlinear jumping stochastic differential equation driven by Poisson random measure and an additional Brownian motion W which is created by the central limit theorem. This Brownian motion is underlying each particle's motion and induces a common noise factor for all neurons in the limit system.
Conditionally on W, the different neurons are independent in the limit system. This is the ``conditional propagation of chaos'' property. We prove the well-posedness of the limit equation by adapting the ideas of Graham 1992 to our frame. To prove the convergence in distribution of the finite system to the limit system, we introduce a new martingale problem that is well suited for our framework. The uniqueness of the limit is deduced from the exchangeability of the underlying system.
This is a joint work with Xavier Erny and Dasha Loukianova, both of university of Evry.
